Contraction Coefficients for Borda Mouthpieces
Wei-guo Dong1 and John H. Lienhard1
Nomenclature
a = distance from a point at
which D j is measured to the
point above it, at which we
want to know Dj,ae,ual
C, C , = arbitrary constants
Cc = coefficient of contraction,
(jet area)/(aperture area)
C, = coefficient of discharge, CcCu
C, = coefficient of velocity,
(jet velocity)/*
D, Dj = diameter of a Borda
mouthpiece, diameter of the
jet from a Borda mouthpiece
g = acceleration of gravity
h = head above a Borda
mouthpiece
M = 4([2L/DI2 + 1)
L = length of a Borda mouthpiece
p = pressure in the liquid
Re = Reynolds number,
p~/(viscosity)
r = radial coordinate around the
axis of the Borda mouthpiece
u = liquid velocity
We = Weber number, pD(2gh)/a
p = liquid density
a = surface tension
Introduction
Figure 1, which is half sketch and half photograph, shows
liquid emerging from a Borda mouthpiece; and it establishes
some nomenclature. It is well-known that an elementary
momentum balance specifies the coefficient of contraction,
C,, (the ratio of the cross-sectional area of the jet to that of
the hole) in the following way:
cC= 1/(2cU2)
(1)
where C, is the "coefficient of velocity" or the ratio of the jet
velocity to
a.
'Heat Transfer/Phase-Change Laboratory, Mechanical Engineering Department. University of Houston. Houston, Texas 77004. Mr. Wei-guo Dong is a
Visiting Scientist on leave from the Thermal Power Engineering Research Institute. Xi-an.. Peo~les'
. Reoublic of China. Professor Lienhard is a Fellow
ASME.
Contributed by the Fluids Engineering Division of THEAMERICAN
SOCIETY
OF
MECHANICAL
ENGINEERS.
Manuscript received by the Fluids Engineering Division, July 9, 1985.
Journal of Fluids Engineering
Good Cc data for the Borda mouthpiece are very hard to
find. The values of 0.5149 (Borda), 0.5547 (Bidone), and
0.5324 (Weisbach) quoted in [1] - without references and
without giving the dimensions o r proportions of the tube - appear to be the bases upon which textbooks normally quote
values between 0.52 and 0.54. It is customary to rationalize a
value of Cc>0.50 using equation (1) with a n assumed C, between 0.98 and 0.96.
While the shortage of Cc data clearly reflects the limited
practical importance of the Borda mouthpiece, two issues in
this matter d o merit our attention. One is that of questioning
the validity of the routinely-taught momentum balance. The
other is that of checking the common implication that
substantial mechanical energy is dissipated in conventional
orifices.
In 1984 Lienhard V and Lienhard (IV) [2] showed that C,
for a sharp-edged orifice is s o close to unity that we can practically forget about any energy dissipation in such discharges.
They also noted that, although they did not calculate the
dissipation in a Borda mouthpiece, it would certainly be even
less than in a sharp edged orifice. Consequently one must look
elsewhere than to C, for the explanation of C,'s greater than
1/2.
The other place where people often look for the cause of the
higher value of Cc is in manufacturing imperfections in the lip
of the mouthpiece. However, one is hard pressed to see how
minor alternations of the lip could affect the momentum
balance calculation in any way2.
We therefore ask how the momentum balance argument
itself could fail. A key assumption in the argument is that the
liquid is stagnant on the support wall (see Fig. 1.). However, it
is clear that there is some liquid movement o n the wall, and
that it must decrease as the length, L, of the mouthpiece is increased. Liquid motion reduces the pressure o n the wall; consequently the jet momentum must increase to compensate for
it. Since the velocity is constant, it is the area of the jet that
must increase.
Our objective is thus to measure Cc as a function of the
ratio L/D. The result should verify or deny two matters that
we have suggested above. 1) If liquid movement parallel with
the support wall affects C,, then Cc should decrease toward an
asymptote as L / D becomes large. 2) If C, is virtually unity,
then that asymptote should be 1/2.
Experiment
Experimental Design. Figure 1 shows the Borda
mouthpiece we designed t o measure C,. The apparatus was
manufactured entirely from plexiglass. The support wall
shown in the figure forms the bottom of 154 cm vertical supply tube with a 15.2 cm inside diameter, which is not shown.
'we are indebted to A. Degani who triggered this inquiry by questioning the
machining argument on this basis, when it was offered in a fluid mechanics
class.
SEPTEMBER 1986, Vol. 1081377
/
In the second experiment, the loss of liquid from the supply
tube was observed as a function of time during draining, and
the discharge coefficient, C,, was obtained from these data.
The supply tube was drained from an initial head, t o each of
about five lower values of head, for each mouthpiece. Each of
these 25 points was replicated about 10 times. The probable error of these measurements averaged 0.83 percent.
We made most of these measurements without letting the
water stand for hours. But when we did let it stand, C, was
the same within 0.4 percent o r less. Nevertheless, these are
time-averaged data a n d , during part of the time in all cases,
the jets exhibited imperfections. Therefore these data are less
trustworthy than the C, observations even though their
probable error is less.
The velocity coefficients, C,,= C,/C,, obtained from the
data did not deviate significantly o r consistently from unity,
which bears out the assertion that C,.= 1 , made in [2].W e
therefore take C, and C,, d a t a as being equivalent.
The ranges of Reynolds and Weber numbers in these tests
were:
23,000 < R e < 62,000 and 800 < W e < 4000
These ranges, which arose entirely from the variability of h ,
had no discernable influence o n C,, o r C,.
Corrections. T h e following corrections were applied to the
observed jet diameters:
1 ) Gravity Correction. Combining the increase of jet
velocity resulting from gravity with the continuity equation we
obtain for the jet diameter uninfluenced by gravity, D,:
D, =(1 +a/2h)"D,,,,,,,,,,
Fig. 1 Borda mouthpiece configuration used in the experiments, and a
typical experimental photograph
The five Borda mouthpieces-each
of a different
length - were separate inserts as shown. Each had a nominal
outside diameter of 1.27 cm. Since plexiglass expands by
about 0.2 percent when it is wet, actual data are based o n the
measured outside diameter of the wet mouthpieces. The lip of
each mouthpiece was examined under a powerful magnifying
glass before it was used, and found t o be free of perceptible
defects.
T a p water was supplied through an overhead tray at the top
of the supply tube. All of the jets filled the Borda tube under
normal conditions. The separated jet flow pattern had to be
initiated by blowing compressed air into the lip region through
a small straw3.
Two kinds of experiment were made. In one, contraction
coefficients were obtained from still photographs of the
discharging jets, by comparing the jet diameter in the picture
with a 1.27 cm marker (which can be seen in the photographic
portion of Fig. 1.) All photographs were made after allowing
the water several hours of stilling to get rid of air bubbles. W e
discovered that air bubbles coming out of solution would cling
to the lip of the mouthpiece and deform the jets. Stilling the
water eliminated most of these bubbles and yielded a higher
fraction of perfectly shaped jets.
About 20 photos were made for each mouthpiece. Just over
half o f these had t o be rejected because of deformities in the
jet. The probable error of C, was 1.3 percent. The standard
error of the repeated measurements varied slightly, since data
were obtained at different heads in each case; but it averaged
1.2 percent which is, predictably, almost the same.
-
' w e are grateful to Prof. T. B. Benjamin for drawing our attention to this
technique, which is needed to operare a Borda mouthpiece.
3781 Vol. 108, SEPTEMBER 1986
(2)
where "a" is the distance from the position at which the
measurement was made to the ~ o s i t o n at which the iet
originated, and h is the head above the mouthpiece. Jets that
did not match equation (2)within the experimental error, over
their length in the photographs, were deemed to be deformed
and were rejected.
2) The Location of the Origin of the Jet. When a
mouthpiece of finite size is located in a gravity field, the formation of the jet is smeared over a variable head region.
Where the effective origin of the jet is located is not clear;
however it should lie between the lip of the mouthpiece and
the support wall.
The gravity correction gave the most consistent results
among the various heads when zero head was taken t o lie one
third of the way from the lip to the support wall. W e therefore
refer the head to this point in each case. T h e variation of the
value of C,, as the zero head point for the correction was
varied from the lip to the support wall, remained well within
the probable error.
3 ) Surface Tension. W e are indebted to T. B. Benjamin
who made the calculation in the Appendix for us. It shows
that the jet dilates slightly under the influence of surface tension. The observed values of C, had to be reduced between 0.4
and 2 percent, on the average, on the basis of this correction.
Results and Discussion
The fully corrected C, and C, data re plotted against L / D
in Fig. 2. (Since they are corrected for surface tension, these
data represent the high W e limit.) They show that C, exceeds
0.5 when L / D is small, and that it falls off asymptotically
toward 0.5 as L / D is increased. The sharp-edged orifice C,, is
included a t L/D=O. (This is Medaugh-Johnson's [ 3 ] value of
0.595 for large We and Re.)
A very simple analytical model serves t o illuminate this
behavior. We approximate the flow on the support wall by imagining symmetrical sinks located at a distance, L from the
Transactions of the ASME
4) The sharp-edged orifice is not the appropriate limit for a
very short Borda mouthpiece.
5) The influence of surface tension on C,. for a Borda jet is
Medaugh-Johnson [3] C D for a s h a r p e d g e d orifice a t high R e a n d W e
0.58
4
I
Cc from draining t e s t s
CD
error b a r s
1
-
-
References
I Encyclopaedia Brilannica, l l th ed., Encylopaedia Britannica Inc., New
York, 191 I, Article on "Hydraul~cs." pp. 38-56.
2 Lienhard, J. H., V and Lienhard, J . H . , (IV), "Velocity Coefficients of
Free Jets from Sharp-Edged Orifices," ASME JOURNAL
OF FLUIDSENGINPERINC, V01. 106. NO. I , 1984, pp. 13-17.
3 Medaugh, F. W . , and Johnson, G. D., "lnvestigalion of [he Discharge
Coefficients o f Small Circular Orifices," Ciuil Engr., Vol. 7 , No. 7. 1940. pp.
422-4.
,
k
m
?
.-
-
5J
-
A P P E N D I X
Approximate prediction,
equation ( 6 )
0.48
I
1
I
I
0
1.o
l
l
1
2.0
i
Influence of Surface Tension o n C , for a Borda Mouthpiece
A Calculation by
T . B. Benjamin, Sedleian Prof. of Natural History
Mathematical Institute
Oxford University
LID
Fig. 2 Experimental results shown the influence of the mouthpiece
configuration on contraction
wall on the jet centerline- that is, where the liquid enters the
tube, and at its mirror image below. The velocity, u, parallel
with the wall is then:
u=-
C
(L'
+ r2)
(4)
where r is the radius in the plane of the wall and C is a constant. Then the pressure gradient on the wall is:
The Bernoulli equation between the stagnant liquid at gauge
pressure, pgh, in the supply tube and the liquid in the jet, is:
pgh = p + pu2/2
('41)
where the static gauge pressure in the jet is given by the
Laplace relation as:
Neglecting fluid motion o n the support wall as being of
second order in importance, we write the following momentum balance:
Force o n static fluid
=
momentum of jet
surface force on the lip
s u r f a c e force o n the jet.
-
where C , must be pghC;D4/16 t o make the velocity consistent
with the discharge rate.
We obtain C, from equation (5) in the following way: W e
first integrate it t o estimate the pressure force o n a large circular segment of the support wall surrounding the hole of
diameter, D . W e then add the jet momentum force,
C,(*D/4)~(2gh), t o get the total force t o the left. This must
be balanced by the opposing pressure within the static liquid,
Solving the result for C, , we obtain:
c,;:M-(fl
-ME
(6)
where M = 4([2L/DI2 + I).
Equation (6) is included in Fig. 2. It is suprisingly close to
the data, considering what a simplistic model it is. It makes it
very clear that fluid movement on the wall is indeed the reason
that observed values of C, are above 1/2.
It is also important to note that neither this simple theory,
nor the data, approach the sharp-edged orifice data point at
the L = O limit. Once the tube is removed entirely, the
transverse velocity component at r = D / 2 discontinuously goes
t o J 2 g h . But for any finite Borda tube, it will be less than this
value.
Conclusions
1) ~ l ~ imotion
d
on the support wall influences the
magnitude of C, for a finite Borda mouthpiece.
2) C, for a ~~~d~ mouthpiece approaches
asymptotically as the length of the mouthpiece is increased. For
a very short Borda mouthpiece' Cc is a little above 0.54'
3, The coefficient
is
a BOrda
mouthpiece, just as it is for a sharp-edged orifice.
Journal of Fluids Engineering
or:
-
T~GDU
aDa
-
(A3)
Substituting equation (A2) in (A[), solving for pu2, and using
theresult in equation (A3), we get a quadratic equation in
x'C, . Its positive root yields:
after we neglect terms of higher than first degree in (l/We).
This result is accordingly limited t o W e greater than the order
of 100. It is also limited t o such We's by the fact that the correction is assumed small enough t o be superposed additively
o n the ideal flow.
Prediction of Stack Plume Downwash
Introduction
Downwash in the wake of smokestacks and cooling towers
has been a subject of investigation for more than 50 years [ I ] .
The phenomenon occurs when the emerging positively
Engtneering Analysis, Inc., Hun~sville,Ala. 35801
Contributed by the Flulds Engineering Divirion o f . T ~ AMERICAN
c
SOCIFTY
OF
MECHANICAL
ENGINEERS.
Mauscript received by the Fluidr Engineering Division
A U ~ U S15,
~ 1985.
SEPTEMBER 1986, Vol. 108 1379